# Fixed effect turns insignificant when including random effect - Multilevel

I have a data set from a diary study in which stress was assessed for 30 days. I want to build multilevel regressions (level 1: measurements, level 2: persons) to investigate the effect of different time-variant predictors on stress. I have a level 1 predictor, which I have within-person-centred, and I am reintroducing the respective person-means on level 2. When I estimate a model with only a fixed effect of my predictor, as such:

fit1 <- lme(fixed = stress ~ 1 + predictor_centred + predictor_mean, random = ~ 1 |ID, data = data, method = "REML", na.action=na.exclude)
summary(fit1)

Value  Std.Error   DF   t-value p-value
(Intercept)        1.9348732 0.10978983 2959 17.623428  0.0000
predictor_centred  0.0128509 0.00569165 2959  2.257845  0.0240
predictor_mean    -0.0076657 0.05816799 2959 -0.131785  0.8952


I get a significant effect of the within-person centred predictor and a nonsignificant effect of the person-means. When I add a random effect of my predictor, a likelihood ratio test indicates that there is significant variation in the slope between persons. The fixed effect of my centred predictor turns insignificant. What does this mean/how do I interpret this and what does it mean for model-building? Thank you in advance!

fit2 <- lme(fixed = stress ~ 1 + predictor_centred + predictor_mean, random = ~ 1 + predictor_centred|ID, data = data, method = "REML", na.action=na.exclude)
anova(fit1, fit2)

Model df      AIC      BIC    logLik   Test  L.Ratio p-value
fit1       1  5 7467.134 7497.275 -3728.567
fit2       2  7 7393.375 7435.572 -3689.688 1 vs 2 77.75881  <.0001

summary(fit2)

Value  Std.Error   DF   t-value p-value
(Intercept)       1.934587  0.10980636 2959 17.618166  0.0000
predictor_centred 0.0058862 0.01121010 2959  0.525081  0.5996
predictor_mean    0.0039029 0.05785693 2959  0.067457  0.9462


## 1 Answer

What random slop do is that the relationship between the outcome and the predictors is now varied by person, every person has their own regression line. What anova output indicates is that allowing this random factor is a better fit for the data. A possible explanation why it became insignificant that in the first model you forced the data to fit with a fixed slop that made the relationship look like significant, but in fact, it is the structure rather than the relationship that is significant.